
// Make newform 324.6.e.c in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_324_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_324_e_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_324_6_e_c();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_324_6_e_c();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        poly := [1, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_324_e();"
function MakeCharacter_324_e()
    N := 324;
    order := 3;
    char_gens := [163, 245];
    v := [3, 1];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_324_e_Hecke();"
function MakeCharacter_324_e_Hecke(Kf)
    N := 324;
    order := 3;
    char_gens := [163, 245];
    char_values := [[1, 0], [0, -1]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..(prec+1)]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 6;
    raw_aps := [[0, 0], [0, 0], [0, 0], [25, -25], [0, 0], [0, 427], [0, 0], [-1711, 0], [0, 0], [0, 0], [0, 10324], [-6661, 0], [0, 0], [3352, -3352], [0, 0], [0, 0], [0, 0], [-56927, 56927], [0, 37939], [0, 0], [79577, 0], [-90857, 90857], [0, 0], [0, 0], [-177725, 177725], [0, 0], [0, 211477], [0, 0], [114482, 0], [0, 0], [-267100, 0], [0, 0], [0, 0], [0, 454657], [0, 0], [127627, -127627], [0, -109214], [-325873, 0], [0, 0], [0, 0], [0, 0], [619001, 0], [0, 0], [0, -656375], [0, 0], [-102493, 0], [0, -1236299], [-304052, 304052], [0, 0], [0, -1269818], [0, 0], [0, 0], [1733551, -1733551], [0, 0], [0, 0], [0, 0], [0, 0], [-365851, 0], [2486614, -2486614], [0, 0], [0, -1952], [0, 0], [-2301136, 0], [0, 0], [3300973, -3300973], [0, 0], [-3692999, 3692999], [0, 4116811], [0, 0], [787957, -787957], [0, 0], [0, 0], [-5160539, 5160539], [0, -1983875], [69893, 0], [0, 0], [0, 0], [5789486, 0], [0, 0], [0, 5170207], [0, 0], [-7111901, 7111901], [0, 0], [1857602, 0], [-3993332, 3993332], [0, 0], [0, 0], [8257450, -8257450], [0, 0], [0, 9039073], [0, 0], [0, 0], [-8305489, 0], [0, 0], [0, 8361568], [0, 0], [0, 0], [0, 0], [-11728675, 0], [13008077, 0], [11304889, -11304889], [0, 0], [0, 0], [0, 0], [0, -9549977], [-12909925, 0], [0, 0], [0, 0], [0, 0], [-6021626, 6021626], [0, 84361], [16390223, 0], [0, 0], [12226039, -12226039], [-1987651, 0], [0, 0], [0, -20670200], [0, 0], [0, 0], [0, 0], [0, -19792751], [22418773, -22418773], [0, 0], [0, 0], [-17363048, 17363048], [0, 0], [935611, -935611], [0, 0], [28451236, -28451236], [0, 11302750], [-29603536, 0], [0, 0], [0, -26183201], [-31143475, 0], [0, 0], [0, 30884593], [0, 0], [0, 25702375], [0, 0], [0, 0], [22770776, 0], [0, 0], [0, 38714323], [0, 0], [-16780807, 0], [0, 0], [-38219123, 38219123], [0, 0], [0, -35829707], [0, 0], [0, 37391125], [0, 0], [16846775, 0], [0, 0], [-26211989, 26211989], [0, 0], [46303868, 0], [0, 0], [5297075, 0], [0, 0], [0, 0], [0, 0], [0, 32345311], [0, 0], [0, 0], [0, 0], [60552827, 0], [44815450, -44815450]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_324_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_324_6_e_c();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_324_6_e_c(:prec:=2)
    chi := MakeCharacter_324_e();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 6));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_324_6_e_c();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_324_6_e_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_324_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<5,R![0, 1]>,<7,R![625, -25, 1]>,<11,R![0, 1]>],Snew);
    return Vf;
end function;